Slater determinant - Aetherwisp

The Slater determinant is a method of determining the wavefunction of a system of many fermions through the determinant of a matrix, in particular the many-electron atom. It is useful because it inherently satisfies the antisymmetry under particle exchange of fermionic states mandated by the Pauli exclusion principle. For a system of $N$ fermions it reads

\Psi(q_{1},q_{2},\ldots,q_{N})=\frac{1}{\sqrt{ N! }}\begin{vmatrix} u_{\alpha}(q_{1}) & u_{\beta}(q_{1}) & \ldots & u_{\nu}(q_{1}) \\ u_{\alpha}(q_{2}) & u_{\beta}(q_{2}) & \ldots & u_{\nu}(q_{2}) \\ \vdots \\ u_{\alpha}(q_{N}) & u_{\beta}(q_{N}) & \ldots & u_{\nu}(q_{N}) \end{vmatrix}

where $q_{1},\ldots,q_{N}$ are the generalized coordinates representing both position and Spin and $u(q)$ are the single-particle wavefunctions for both position and spin. The subscripts $\alpha,\beta, \ldots,\nu$ denote the sets of quantum numbers $(n,l,m_{l},m_{s})$ that uniquely determine the simultaneous space and spin state of the fermion.

Properties#

The usage of a determinant enforces antisymmetry and the exclusion principle: if any two $u_{i}$ and $u_{j}$ are equal, or if they are exchanged, the set $\{ u_{i} \}$ becomes linearly dependent, the rank of the matrix is no longer maximum and the whole determinant, which is the wavefunction, becomes a constant zero.